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The field of Nash functions and factorization of polynomials. (English) Zbl 0909.12002

A holomorphic function \(f:U\to \mathbb{C}\) defined on an open connected subset of \(\mathbb{C}^m\) is a Nash function if there is an irreducible polynomial \(P(X,Z) \in\mathbb{C} [X_1, \dots, X_m,Z]\) such that \(P(u,f(u))=0\) for every \(u\in U\). The Nash functions defined on \(U\) form an integral domain, \(N_U\). If \(V\subseteq U\) is another open connected set, then the restriction defines a homomorphism \(N_U\to N_V\). Using a certain filter base of open connected sets, the author studies the direct limit \(N=\lim_\to N_U\). He proves that \(N\) is an algebraically closed field, hence an algebraic closure of the function field \(\mathbb{C}(X_1, \dots, X_m)\). The factorization of polynomials \(P\in\mathbb{C} [X_1, \dots, X_m;Y_1, \dots,Y_n]\) into irreducible factors of \(N[Y_1, \dots, Y_n]\) is studied.

MSC:

12D05 Polynomials in real and complex fields: factorization
12F99 Field extensions
32A10 Holomorphic functions of several complex variables
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